Acoustics & Music Theory
why it sounds like that
The collection

Every essay — page 24

Page 24 of 27, continuing through the fields in the same order.

Pitch and tuning Intervals and chords Scales and modes Harmony and voice leading Rhythm and metre Timbre and acoustics Perception and the listener Instruments and their design Form and structure Series Objects Sounds Search

Instruments and their design

Where the sound came from before any of the above. A stopped tube has only odd partials, a hammer at one seventh silences the seventh, and a bass string is wound because a plain one would be longer than the room.

Second order and fourth order, from the same sequence of wavenumbers. Frequency against wavenumber for a string and for a bar, with each object's own wavenumbers marked on the axis. A string's equation is second order in space, so frequency goes as wavenumber and the straight line carries its modes at 1, 2, 3, 4. A bar's is fourth order, so frequency goes as the SQUARE of wavenumber and the parabola carries the same kind of sequence to 1, 2.757, 5.404, 8.933. Both sequences of wavenumbers are arithmetic — a string's are the even multiples of π/2L and a bar's the odd ones, except that the first is 3.011 rather than 3 — and squaring an arithmetic sequence is the whole of why one object has a fundamental and the other has none. Every number is derived from the boundary condition; none is measured.

A bar's partials are the odd numbers, squared

A string's wave equation is second order in space and a bar's is fourth, so a string's frequencies go as its wavenumbers and a bar's go as their squares. Both objects fit a nearly arithmetic sequence of wavenumbers between their ends; squaring one is the whole of why a marimba key has no note in it. The ratios are 1 : 2.756 : 5.404 : 8.933, which are the odd squares over nine, uniformly 12.9 cents flat, and the 12.9 cents is one root that refused to be 3π/2.

7 figures Where one arch can put a bar's second and third partials. The plane of the second and third partials, with the path a single arch traces through it as it is cut deeper, for 4 arch lengths from 50 to 100 per cent of the bar. Every path starts at the plain bar's 2.756 and 5.404 and runs up and to the right. The xylophone's target of 3 and 6 lies on them: reach the second partial and the third is there. The marimba's target of 4 and 10 lies above every one of them — the best any of these arches manages while holding the second partial at 4 is 9.176, which is 149 cents short. A marimba's third partial is not something one cut can place, and that is a statement about the shape of the cut rather than about the maker.

One cut cannot place two partials

An arch cut into a bar's underside is one knob, and a maker aims it at two targets. Swept across every arch length, the path it traces through the plane of the second and third partials passes through the xylophone's 3 and 6 — dead through it, within a tenth of a cent at the natural arch length — and under the marimba's 4 and 10 by between 149 and 326 cents. Hitting one target means missing the other, and that is why a marimba bar's underside is not a simple arch.

7 figures What a stopped tube tuned to the fundamental does to a marimba bar's partials. Power radiated from the mouth of a 311 millimetre stopped tube of 56 millimetre bore, tuned so that its first resonance is the bar's fundamental at 262 hertz, plotted against multiples of that fundamental and referred to what it does at the fundamental itself. Its quality factor there is 80. A stopped tube resonates at the odd multiples and presents an infinite input impedance — a rigid lid — at the even ones, so a partial's fate is decided by the parity of the ratio the arch put it on. The bar's own partials are marked: 1 at 0.0 decibels, 4.00 at -38.5 decibels, 9.03 at -13.1 decibels.

The tube shuts on the partial the arch placed

A stopped tube tuned to a bar's fundamental resonates at every odd multiple of it and presents a rigid lid — an infinite input impedance — at every even one. A xylophone's arch puts its second partial on 3, which is a resonance, and the tube passes it within five decibels. A marimba's puts it on 4, which is an antiresonance, and the tube takes it thirty-eight decibels down. Same tube, opposite answers, and the difference is parity.

7 figures What each strike point on a marimba bar excites. The amplitude each of the first 4 partials receives from a strike at each point along the bar, each partial drawn to its own maximum. On a string this comb is |sin nπx| and is periodic, so one strike point silences a whole family; on a bar it is the mode shape itself, whose zeroes are not at rational fractions, so no strike point silences a family. The exception is the middle: every antisymmetric mode is zero there by symmetry, so a centre strike drives partial 1 at 100 per cent of its maximum and partial 2 at nothing at all. The cord's hole at 0.2031 is the mirror case — the fundamental is the partial a strike there cannot reach.

The four ways a marimba loses what its arch placed

The centre of a bar is a node of every even mode by symmetry, so a stroke there gives the second partial exactly nothing and a mallet's width cannot recover it; five millimetres off centre returns 14 per cent. Add the yarn mallet's 3-millisecond contact, the tube's rigid lid and a decay four times faster than the fundamental's, and four independent mechanisms take the tuned partial 80 decibels below it. On a xylophone the same four come to 24.

7 figures Partials 3, 4, 12 of "hod", over one vibrato cycle. The level of three partials of a 220 hertz note on the vowel in "hod", each about its own mean, over one cycle of a vibrato of ±71 cents at 6.0 hertz. The pale curve is the frequency deviation itself, for phase reference. Partial 3 at 660 hertz swings 4.22 decibels and peaks with the frequency; Partial 4 at 880 hertz swings 0.41 decibels and peaks twice a cycle; Partial 12 at 2640 hertz swings 8.62 decibels and peaks against it. The formants of this vowel are at 730, 1090, 2440 hertz and do not move; a partial below one rises as the frequency rises and one above it falls, so the modulations of a single note run in opposite directions at the same instant.

The partial that gets louder as it goes sharp

Eleven earlier essays sweep a set of partials and hold their amplitudes still, and a real tract does not move with the fundamental. Put the formant account and the sweeping account together and every partial acquires an amplitude modulation at the vibrato rate: 0.07 decibels on the fundamental and 8.6 on the twelfth partial of the same note. They are in phase below a formant and anti-phase above one — not ninety degrees apart — and the whole note swings 0.82 decibels, because they cancel.

7 figures Where a section's fluctuation stops being a beat, on 220 hertz. Two rates up the spectrum of a 220 hertz note sung by a section whose voices are spread by 15 cents. The rising line is the beat rate between a typical pair of them, which grows with the partial because a mistuning in cents is a difference in hertz that scales with frequency; it reaches the 15 hertz at which a beat stops being a beat by partial 5.5, at 1217 hertz. The flat line is the amplitude modulation the vibrato imposes through the formants, which is 6.0 hertz at every partial because the vibrato modulates every partial by the same number of cents at the same rate. The two are equal at 487 hertz. Above 1217 hertz the beating has become roughness and the only fluctuation left is the vibrato's — and that frequency is the same one an octave up, where it is partial 2.8 instead.

The rate that does not rise with the partial

Twelve earlier essays give every vibrato the same six hertz, and the measured spread is 5.5 to 7.5. Putting the two fluctuations a choir contains on one axis shows why the rate matters: the beating between mistuned voices rises with the partial and leaves the range a listener follows as fluctuation at 1,217 hertz, while the vibrato's own modulation is six hertz at every partial. Above that frequency a section fluctuates by vibrato alone — and if every singer had the same rate, it would barely fluctuate at all.

7 figures The partials the air makes, on the series the bore actually has. The input impedance of a trumpet, with the partials wave steepening manufactures from its A4 at 428 hertz drawn on it as vertical marks. The steepening is a distortion of one periodic waveform, so it makes its partials at exact integer multiples — 856, 1285, 1713, 2141, 2569 hertz. The bore's own resonances are not at integer multiples of anything: the peaks the manufactured partials aim at sit at 886, 1346, 1812, 2278, 2741. So every manufactured partial lands flat of the peak it might have used, by 59, 81, 98, 108, 112 cents — 2.6, 2.7, 3.0, 4.0, 4.9 half-widths of the peaks in question, which is outside the half-power point of every one of them. The dashed line is where the bell stops reflecting, at 2945 hertz; above it there is no peak to land on or miss.

Which notes go brassy first

Wave steepening manufactures partials at exact integer multiples of the note being played. A brass instrument's resonances are not at integer multiples of anything, and twelve earlier essays have measured how far off they are without ever putting the two on one axis. Put them there and a manufactured partial never lands on a peak — never once, at any note, by a miss that is the same number of hertz every time. So the alignment cannot be what decides which notes blare, and the thing that does turns out to be the bell.

7 figures Four throats on one instrument, drawn to scale. 4 bores 148 centimetres long, drawn to scale in radius and length, differing only in the radius of the tube itself — 6.0, 9.6, 14.0, 22.0 millimetres across at the throat. The mouth is held at 124 millimetres, which is what a maker changing a leadpipe actually does. A consequence is that the flare ratio falls from 20.7 to 5.6 and the bell's own boundary with it, from 5906 hertz to 1158. Beside each is the fourth resonance and how sharp it is: 427 Hz at Q 23, 428 Hz at Q 34, 427 Hz at Q 44, 421 Hz at Q 51. The notes move by 27 cents across the whole sweep and the sharpness rises by a factor of 2.2.

The throat that decides both

Eight earlier essays draw a tube eleven millimetres across and none of them draws another. Opened from six millimetres to twenty-two, it does exactly what was predicted to the loss in the walls — and it moves the radiation loss three times further, in the opposite direction, and monotonically, which the bell does not. So the two ends of a brass instrument do not divide the two losses between them. One end owns one loss and both ends own the other.

7 figures Two registers 50 microseconds apart on one string. The partials of a 355-millimetre string plucked at 32 and 50 millimetres from the nut, drawn twice: once with the two quills releasing together, once with the far one releasing 0.05 milliseconds later, which is 0.026 of this string's period. A delayed release turns partial n through 2·pi·f_n·dt, so the rotation is proportional to the partial number: the fundamental is turned 9 degrees and partial 24 is turned 226. Simultaneous, the pair is missing partials 17 and 20 — holes it digs for itself where the two combs are equal and opposite. Staggered, it is missing none of them: a rotation of anything at all takes two amplitudes out of opposition. The partials neither comb can excite at all — 7, 11, 22 — are filled either way, because where one comb is zero the sum is the other one whatever its phase. The fundamental's gain over the far register alone falls from 4.36 decibels to 4.34.

The interval between two quills

Two jacks on one key are voiced separately and do not let go at the same instant. That interval turns each partial of the later pluck through an angle proportional to its number — so it leaves the fundamental alone and inverts the twentieth partial, and the holes the pair digs for itself vanish at a hundredth of a period. What a regulator can tolerate turns out to be one fixed fraction of a period at every pitch, which on a five-octave instrument is a factor of sixteen in milliseconds.

7 figures Where a bar and its pipe stop being two things. The two normal modes of a bar and a resonator tuned to it, against how strongly they are coupled, at 262 hertz. Below a threshold the pair has one frequency and two different decay rates — the pale curves, which are the damping splitting rather than the pitch — and above it the frequencies separate. The threshold is exact and it is not a matter of degree: it is where the coupling rate equals half the difference between the two damping rates, which for a bar of Q 197 against a tube of Q 80 is a coupling of 0.37 per cent. A marimba's own coupling is 0.62 per cent — 1.67 times the threshold, and not free: it is fixed by how much louder the tube makes the note, since the coupling that splits the pair is the coupling that carries the energy out. So the resonator model's assumption that the tube is a filter downstream of the bar is wrong at middle C, and it is wrong by less than a factor of two.

A bar and its pipe are one object

Three earlier essays treat a marimba's resonator as a filter the bar's output passes through, and both of them said in their own caveats that the coupling was not modelled. It is here, and the debt was right: the coupling is 1.67 times the threshold at which the pair acquires two frequencies instead of two decay rates, so the tube is not downstream of anything. Every consequence of that is smaller than the peaks it would have to be seen between.

7 figures Where a woodwind's A♭3 leaves it, below its corner and above it. The same fingering — 6 holes open on a 15-millimetre bore 567 millimetres long, sounding A♭3 at 207 hertz — drawn twice, with each opening's circle scaled by the share of the radiated power that leaves through it. At 400 hertz 78 per cent of it leaves through the first open hole, the bell takes 0 per cent, and the number of apertures really doing the radiating is 1.6; At 2600 hertz 6 per cent of it leaves through the first open hole, the bell takes 49 per cent, and the number of apertures really doing the radiating is 3.3. The lower frequency is below this fingering's corner and the higher one above it: below the corner the instrument is a short tube with one opening at the end of it, and above the corner it is the whole lattice at once. The power-weighted station — where a listener would say the sound is coming from — moves from 392 millimetres to 515.

Where a woodwind actually sounds from

Every number so far is read at the mouthpiece, and the corner's whole musical meaning is at the other end. Run the same solver forwards and it gives the flow leaving every hole — from which a clarinet's radiating aperture turns out to be a function of fingering and of frequency, but not the way it was predicted to: the fingering sets how far the aperture opens, almost exactly to the number of open holes, and barely moves the frequency at which it does.

7 figures One doubling, held down a phrase. Where a single held arrangement of 4 players on 3 notes stands among the 36 at each chord of a 5-chord passage, best at the top, with what each chord would rather have named along the bottom. The held answer is flue pipe · trumpet · clarinet+violin, and it is the chord's own first choice at 4 of 5 of them. Holding it costs 16.2 per cent of the passage's roughness against re-scoring every chord — which is 2.4 per cent of the range the choice actually spans, since the arrangements at one chord differ by a factor of 7.8 on average. The cost is not spread over the passage: 1 chord carries nearly all of it.

An orchestrator doubles a line, not a chord

Three earlier essays made the objective a functional over a passage and a later one went back to holding one chord still. Put the doubling back into time and the retreat turns out to have been cheap: one arrangement held down a five-chord phrase is that phrase's own best answer at four of its five chords and costs 2.4 per cent of the range the choice spans — while the forward mask named earlier as the third temporal constant reaches for twenty milliseconds rather than two hundred, and cannot change the answer at any pace at all.

7 figures The C♯4 tube speaks in C4's attack. A marimba bar at C4, 262 hertz, struck, with the bar a semitone above it next to it in the same row. One curve is the struck note as the tubes radiate it; the other is what radiates at the neighbour's own pitch, C♯4; both in decibels against the struck note's peak. The dashed curve is where the neighbour would have to be to modulate the note by the beat-detection threshold at 15.6 beats a second, 3.1 per cent. The neighbour is above it for the first 90 milliseconds — 1.4 beats — and then drops away, while the note itself is near its loudest. Over the whole note the energy at the neighbour's pitch is 33.8 decibels below the struck note's. It comes out of the neighbour's tube, which the bar reaches at -13.5 decibels, and it leaves because that tube hands it to its own bar, which cannot radiate it.

The second row of resonators keeps the neighbours out of the attack

A marimba bar reaches the mouth of the tube under the next bar at a fifth of what it puts on its own, and the next bar is tuned a semitone away. Solved as a row of coupled pairs, the neighbour's pitch speaks in every attack for about a beat and a half and is then handed to its own bar, which cannot radiate it. Laying the accidentals out in a second bank halves that for eight notes of the twelve, and leaves C, E, F and B exactly where they were.

7 figures A cone opens its aperture as far as a cylinder does only if its holes grow with it. For every fingering of four woodwinds with twelve holes each, each tuned to its own semitones — a clarinet's cylinder with graded holes and with holes all 9 millimetres across, a conical bore of an oboe's proportions with holes 0.6 of the local bore radius, and the same cone with every hole 7 millimetres across — the largest number of openings really radiating together at any frequency above 700 hertz, as a participation number counting the bell. The dotted line is every open hole and the bell sharing equally. clarinet: a cylinder: 1 open 2.0, 2 open 3.0, 3 open 3.9, 4 open 4.8, 5 open 5.6, 6 open 6.5, 7 open 7.2, 8 open 7.8, 9 open 8.8, 10 open 9.6, 11 open 10.1, 12 open 10.5. cone, holes growing with the bore: 1 open 2.0, 2 open 2.9, 3 open 3.9, 4 open 4.9, 5 open 5.9, 6 open 6.7, 7 open 6.6, 8 open 7.2, 9 open 7.7, 10 open 7.4, 11 open 8.4, 12 open 8.9. cone, every hole 7 mm: 1 open 2.0, 2 open 2.8, 3 open 3.7, 4 open 4.5, 5 open 5.4, 6 open 5.6, 7 open 6.1, 8 open 5.6, 9 open 6.5, 10 open 6.7, 11 open 7.1, 12 open 6.8. clarinet, every hole 9 mm: 1 open 2.0, 2 open 2.9, 3 open 3.9, 4 open 4.9, 5 open 5.9, 6 open 6.8, 7 open 7.7, 8 open 8.6, 9 open 9.6, 10 open 10.1, 11 open 9.2, 12 open 10.2.

A cone radiates like a cylinder only if its holes grow

The prediction was that a conical woodwind radiates from fewer of its holes than a clarinet, because a hole near the wide end of a cone sits on a tube that is already easy to push air along, and so pulls less of the passing flow out. The mechanism is right and the conclusion is wrong for the instruments that exist. With holes that grow with the bore, as an oboe's and a saxophone's do, a cone opens its aperture as far as a cylinder up to six open holes and falls short above that by a hole or two. Only a cone whose holes are all one size tops out far below — near seven apertures with thirteen open. What a cone really changes is where and how much: its aperture opens nearly a major third lower, its bell carries twice the share, and it gets out at best under half of what reaches it, where the clarinet gets out seven tenths.

6 figures A leadpipe, and a cylinder of its own mean bore. The first 60 centimetres of a 148-centimetre trumpet bore, with the radius drawn above a floor of 2.5 millimetres and magnified, so that a millimetre can be seen. The top row is the bore every earlier figure drew: 11.0 millimetres across from the mouthpiece to the flare. Below it are three leadpipes opening from 8.8 millimetres at the receiver to the bore, over 10, 25, 40 centimetres, each drawn over the dashed outline of a cylinder of the taper's own mean radius. At the right is how far each set of resonances sits from a harmonic series, in cents: no leadpipe 31; 10 cm, taper 22 against cylinder 21; 25 cm, taper 15 against cylinder 24; 40 cm, taper 14 against cylinder 33.

A leadpipe tunes the series, and its average does not

Every trumpet drawn in the essays before this one runs at full bore from the mouthpiece to the flare, and no trumpet does. Put a real taper in front of it and the wall losses it was predicted to cost come to a fraction of a per cent. What it does instead is pull the resonances onto a harmonic series, halving their average mistuning, while a cylinder of the same mean bore makes them worse.

8 figures A vibrato hides a section's beats above about four hundred hertz. Two voices of a section whose tuning is spread by 15 cents, singing one note: how many nulls of their beating a listener can follow each second — counted only while the beat is at or below fifteen a second — at the frequency of the partial compared, sung straight and with a 71-cent vibrato at about six hertz. 220 Hz 2.2 straight, 5.3 with vibrato; 330 Hz 3.3 straight, 4.4 with vibrato; 440 Hz 4.2 straight, 3.9 with vibrato; 550 Hz 5.0 straight, 3.3 with vibrato; 660 Hz 5.1 straight, 2.8 with vibrato; 880 Hz 5.5 straight, 2.4 with vibrato; 1100 Hz 4.8 straight, 2.0 with vibrato; 1320 Hz 4.6 straight, 1.5 with vibrato; 1760 Hz 3.9 straight, 1.2 with vibrato; 2200 Hz 3.5 straight, 1.1 with vibrato; 2640 Hz 3.2 straight, 0.9 with vibrato; 3200 Hz 2.3 straight, 0.7 with vibrato; 3520 Hz 2.0 straight, 0.7 with vibrato. Below 413 hertz the vibrato adds followable beats, because it sweeps a slow difference faster. Above it the vibrato removes them, because it carries the pair's difference past fifteen hertz most of the time; at 3,200 hertz, the singer's formant, a straight section gives 3.4 times as many.

A vibrato hides a choir's tuning above four hundred hertz

Thirteen essays drew a vibrato that had always been running; a real one grows out of a straight tone over the first quarter-second of a note. Counting the beats a listener can follow between two voices of a section shows what that front edge is worth. A vibrato takes most of a section's countable beats away above about 400 hertz — seven in ten of them at the singer's formant — and adds some below. The straight start gives them back only while the vibrato is still growing, and in the band where a voice carries that is less than one beat.

6 figures A section's vibratos must lock almost completely before its tuning is exposed again. Two voices of a section spread by 15 cents, each with a 71-cent vibrato near six hertz, their vibratos pulled together so that their differences of rate and phase shrink to a stated share: followable beats a second at three partial frequencies, against how far they are locked, with the same voices sung straight as the dashed line beside each. 660 Hz (straight 5.14): 0 2.77, 0.25 2.90, 0.5 3.23, 0.7 3.85, 0.8 4.45, 0.85 4.98, 0.9 5.02, 0.95 5.31, 0.98 5.17, 1 5.15; 1217 Hz (straight 4.66): 0 1.60, 0.25 1.80, 0.5 2.04, 0.7 2.58, 0.8 2.88, 0.85 3.29, 0.9 3.88, 0.95 4.41, 0.98 4.67, 1 4.66; 3200 Hz (straight 2.33): 0 0.68, 0.25 0.70, 0.5 0.80, 0.7 1.13, 0.8 1.10, 0.85 1.45, 0.9 1.70, 0.95 1.98, 0.98 2.12, 1 2.23. At the singer's formant, half-locked vibratos expose 34 per cent of what a straight section does, locked to nine-tenths 73 per cent, locked to ninety-five 85 per cent; only at about ninety-eight per cent does it come within a tenth of a straight section.

A choir's vibratos must lock to two milliseconds

A vibrato hides a section's mistuning from a listener counting beats, and the obvious way it could stop hiding it is for the singers' vibratos to move together. They would have to move very nearly together. Half-entrained vibratos leave a section as hidden at the singer's formant as independent ones; locked to nine-tenths they expose three-quarters of what a straight section does; only past nineteen parts in twenty does the section's tuning show as if they sang straight. At six cycles a second that is two vibratos peaking within two milliseconds of each other — and the better a section is tuned, the tighter the lock has to be.

7 figures Heard a little late, two vibratos lock in step; heard a third of a cycle late, they lock in opposition. Two singers' vibratos at 6.0 and 6.2 a second, each pulled towards the phase it hears from the other with a strength of 10 radians a second, the hearing delayed by a stated time: how far apart the two vibratos' peaks settle, in milliseconds, after twenty seconds. 0 ms: 1.6 apart (4°); 20 ms: 2.4 apart (5°); 40 ms: 61.9 apart (-169°); 60 ms: 70.7 apart (-176°); 80 ms: 79.5 apart (-176°); 100 ms: 88.1 apart (-176°); 120 ms: 95.7 apart (-175°); 140 ms: 1.6 apart (4°); 160 ms: 1.6 apart (4°); 180 ms: 1.8 apart (4°); 200 ms: 69.6 apart (-176°). Under about forty milliseconds the lock is near in phase; from about forty to a hundred and twenty it is near antiphase, the peaks most of half a cycle apart; from about a hundred and forty, nearly a whole cycle late, it is near in phase again. The shaded band is the two milliseconds the formant asked for.

Singers who hear each other late lock their vibratos in opposition

A choir's vibratos give back the section's tuning at the singers' formant only if they are locked to within about two milliseconds. Singers who lock do it by hearing each other, and they hear each other late. Two vibratos pulled towards what each hears of the other settle near in step while the delay is under about forty milliseconds, and even then come within two milliseconds only if each singer corrects a phase error in about thirty. From forty to a hundred and twenty milliseconds late — a third to two-thirds of a vibrato cycle — they lock in opposition, peaks seventy to ninety milliseconds apart, and hide the section's tuning more completely than vibratos that never locked at all.

6 figures Where a pair would lock in opposition, a section of sixteen does not lock at all. Sixteen singers in two rows of eight, sixty centimetres apart, each hearing every other after the sound's travel and a common latency, each vibrato pulled towards all it hears with a total strength of 10 radians a second; four draws of rates and phases. Against the latency: R1, the length of the mean of the vibratos' phases, one when every vibrato is in step; and R2, the same for twice the phases, one also when the section has split into two opposed halves. 0 ms: 0.96, 0.86; 10 ms: 0.94, 0.83; 20 ms: 0.89, 0.71; 30 ms: 0.72, 0.52; 40 ms: 0.47, 0.31; 50 ms: 0.09, 0.24; 60 ms: 0.07, 0.22; 70 ms: 0.08, 0.23; 80 ms: 0.08, 0.22; 90 ms: 0.10, 0.24; 100 ms: 0.19, 0.22; 110 ms: 0.68, 0.46; 120 ms: 0.90, 0.73; 135 ms: 0.96, 0.85; 150 ms: 0.97, 0.87. A section split in two would show R2 high with R1 low; it never does.

Sixteen singers cannot all oppose each other

Two singers who hear each other a third of a vibrato cycle late lock their vibratos in opposition and hide their tuning better than if they had not listened at all. Sixteen cannot do it, because no arrangement puts every one of them opposite every other. At the latencies where a pair opposes, a section of sixteen does not lock at all: each singer keeps a vibrato rate of its own, and a listener hears the section's tuning exactly as through independent vibratos. Outside that band the section locks in step and gives most of its tuning back. A section that listens to itself can expose its tuning or leave it as it was, but never hide it.

6 figures A venturi four centimetres long moves the middle of the series as much as the top. The trumpet with a 25-centimetre leadpipe, and the same leadpipe with a venturi four centimetres long narrowing its radius by 20 per cent at its centre, three centimetres from the receiver: how far each of the first ten resonances moves, in cents. The written middle is shaded. 1: +7.3, 2: +6.1, 3: +2.4, 4: -2.4, 5: -7.9, 6: -8.5, 7: -4.2, 8: +1.8, 9: +6.1, 10: +6.7. The largest shift in the written middle is 8.5 cents and above it 6.7; the series' mistuning goes from 15.0 to 14.2 cents rms. The lowest resonances rise, the middle ones fall, and the top rises again.

The venturi is heard by every resonance

The narrowing makers put just inside a trumpet's leadpipe was predicted to be invisible to the written middle and to act only at the top of the compass, because short tapers had acted only there. It is not a taper. To every resonance in the compass a venturi a few centimetres long is one lumped object, described by a single number — its length times its depth — and by where it sits. Four centimetres at a fifth of the radius moves the middle notes by up to eight and a half cents, raising some and lowering others, while the series as a whole stays as well tuned as before.

6 figures A row that hears its neighbours late hides the tuning between them, exposes it between every second singer, and leaves the whole as it was. Sixteen singers in one row sixty centimetres apart, a draw of fifteen cents apart with 71-cent vibratos, each pulled at ten radians a second towards what it hears, each voice's pull weighted by the inverse of its distance to the power 8; four draws. The beats a second a listener can follow at 3,200 hertz, against the latency, for neighbouring pairs, for pairs two places apart, and over every pair. neighbours: 0 ms 1.62, 20 ms 1.36, 40 ms 0.69, 60 ms 0.30, 80 ms 0.28, 100 ms 0.29, 120 ms 0.78, 150 ms 1.63; every second singer: 0 ms 1.23, 20 ms 0.90, 40 ms 0.82, 60 ms 0.94, 80 ms 1.21, 100 ms 1.15, 120 ms 0.90, 150 ms 1.21; every pair: 0 ms 0.96, 20 ms 0.77, 40 ms 0.72, 60 ms 0.67, 80 ms 0.67, 100 ms 0.69, 120 ms 0.75, 150 ms 0.92. Sung straight 2.33, independent vibratos 0.68, a pair in opposition 0.19.

A row that hears its neighbours moves its tuning, it does not hide it

Sixteen singers who all hear each other alike cannot lock in opposition, because opposition between every pair is impossible. A row that hears mainly its neighbours has no such obstacle — opposition along a chain is consistent — and at the latencies that make a pair oppose, it half-alternates: neighbours opposite each other, every second singer nearly in step. That hides the tuning between neighbours as well as a duet hides it, and exposes it between every second singer more than independent voices would. Over the whole row the two cancel to the hundredth. With no latency, the same local hearing stops the row locking at all, and hides nearly half of what a whole-row lock would show.

6 figures Two strings a few cents apart carry the pair through every interval the regulator feared, and its brightness breathes. A harpsichord's two eight-foot registers, plucking at 32 and 50 millimetres from the nut of a 355-millimetre string, each on its own string of a unison at C5, the second tuned 2 cents above the first: the spectral centroid of the pair, in partial numbers, through the first three seconds of a note (both strings decaying alike). At the pluck it is the aligned pair's 2.44; through the note it ranges from 2.29 to 3.67, repeating every 1.65 seconds. The detuning carries the pair past the regulation tolerance of a decibel at the fundamental — 0.153 of a period — after 254 milliseconds (the line). Every half second: 0.0 s 2.44, 0.5 s 3.12, 1.0 s 3.34, 1.5 s 2.53, 2.0 s 2.90, 2.5 s 3.66, 3.0 s 2.89.

Two strings undo what two quills were regulated for

A harpsichord's two eight-foot registers each pluck their own string of a unison, and the unison is never exact. Two cents apart, the second string's partials turn against the first's at a rate proportional to their number, so within a quarter of a second at C5 the pair has drifted past the regulation tolerance the jacks were voiced to, and through the rest of the note it passes again and again through every interval a regulator could have set. The pair's fixed spectrum becomes one that breathes, its own holes become flickers a few per cent of a cycle long, and averaged over the note the second register is worth a decibel and a half at the fundamental, not four.

5 figures At speed each note of a clarinet's rising scale is cut off nearer its steady level than the last, which is a crescendo nobody wrote. A two-octave scale from D3 on a modelled clarinet at 16 notes a second, 62.5 ms a note, each fingering change handing the air column a new resonance that starts from nothing and rises with a time constant set by its Q. The level each note reaches before the next begins: -3.7 dB, -3.4 dB, -3.1 dB, -3.0 dB, -2.7 dB, -2.5 dB, -2.2 dB, -2.1 dB, -1.9 dB, -1.7 dB, -1.6 dB, -1.4 dB, -1.2 dB, -1.0 dB, -0.9 dB. From bottom to top that is a rise of 2.8 dB.

A fast run is a crescendo nobody wrote

A wind note takes tens of milliseconds to build, and a fast run gives each note about sixty. The question sounds like whether a note can arrive at all, and the settling time makes it sound worse than it is, because a note cut off at nine tenths has lost under a decibel. What speed really costs is evenness: the settling time falls up the compass, so at sixteen notes a second a rising two-octave scale gains between two and four and a half decibels with no change in the player's breath. A repeated note, which keeps its resonance, loses only a quarter as much.

6 figures A vibraphone's motor makes a note quieter and far longer, with a tremolo that is shallower than the valve's two states at middle C. The power a struck bar and its tube radiate at middle C, in decibels, with the coupling switched at 6 openings a second. A tube's filling time here is 98 ms. With the bar holding the same energy, the open valve radiates 13.8 dB more than the shut one; with the motor running the tremolo's depth is 10.0 dB.

A vibraphone's valve is a door on the sound and a drain on the bar

A vibraphone's motor is usually described as a duty cycle: loud while the valve is open and quiet while it is shut, with a tremolo as deep as the gap between them. Put the tube's own time back and the description comes apart. The door and the drain obey different laws. A low note's tremolo is a third shallower than a high note's at the same speed, the note rings almost three times longer than with the valve left open and well beyond any mean of the two states, and the same stroke can land eight decibels apart according to where the motor happens to be.

6 figures